If \(\mathbb {E}|X_1|^{2m}<\infty \)  holds for some \(m\in \mathbb {N}\) , \(m\ge 3\) , by [1]}, for \(n\)  sufficiently large, the density of \(n^{1/2}(\overline{Y}-\mu _Y)\) , \(g_n\)  exists, is bounded and admits the expansion
\(\sup _{y\in \mathbb {R}^2} (1+\Vert y\Vert )^{m}\Big |g_n(y)-\sum _{r=0}^{m-2}n^{-r/2}\pi _r(y)\phi _{\Sigma }(y)\Big | ={\begin{matrix}\!{\mathcal {O}}\!\end{matrix}}\big (n^{-(m-2)/2}\big ),\) 
